Testing biochemistry revisited: how in vivo metabolism can be understood from in vitro enzyme kinetics

A decade ago, a team of biochemists including two of us, modeled yeast glycolysis and showed that one of the most studied biochemical pathways could not be quite understood in terms of the kinetic properties of the constituent enzymes as measured in cell extract. Moreover, when the same model was later applied to different experimental steady-state conditions, it often exhibited unrestrained metabolite accumulation.Here we resolve this issue by showing that the results of such ab initio modeling are improved substantially by (i) including appropriate allosteric regulation and (ii) measuring the enzyme kinetic parameters under conditions that resemble the intracellular environment. The following modifications proved crucial: (i) implementation of allosteric regulation of hexokinase and pyruvate kinase, (ii) implementation of V(max) values measured under conditions that resembled the yeast cytosol, and (iii) redetermination of the kinetic parameters of glyceraldehyde-3-phosphate dehydrogenase under physiological conditions.Model predictions and experiments were compared under five different conditions of yeast growth and starvation. When either the original model was used (which lacked important allosteric regulation), or the enzyme parameters were measured under conditions that were, as usual, optimal for high enzyme activity, fructose 1,6-bisphosphate and some other glycolytic intermediates tended to accumulate to unrealistically high concentrations. Combining all adjustments yielded an accurate correspondence between model and experiments for all five steady-state and dynamic conditions. This enhances our understanding of in vivo metabolism in terms of in vitro biochemistry.     

Understanding glucose transport by the bacterial phosphoenolpyruvate:glycose phosphotransferase system on the basis of kinetic measurements in vitro

The kinetic parameters in vitro of the components of the phosphoenolpyruvate:glycose phosphotransferase system (PTS) in enteric bacteria were collected. To address the issue of whether the behavior in vivo of the PTS can be understood in terms of these enzyme kinetics, a detailed kinetic model was constructed. Each overall phosphotransfer reaction was separated into two elementary reactions, the first entailing association of the phosphoryl donor and acceptor into a complex and the second entailing dissociation of the complex into dephosphorylated donor and phosphorylated acceptor. Literature data on the K(m) values and association constants of PTS proteins for their substrates, as well as equilibrium and rate constants for the overall phosphotransfer reactions, were related to the rate constants of the elementary steps in a set of equations; the rate constants could be calculated by solving these equations simultaneously. No kinetic parameters were fitted. As calculated by the model, the kinetic parameter values in vitro could describe experimental results in vivo when varying each of the PTS protein concentrations individually while keeping the other protein concentrations constant. Using the same kinetic constants, but adjusting the protein concentrations in the model to those present in cell-free extracts, the model could reproduce experiments in vitro analyzing the dependence of the flux on the total PTS protein concentration. For modeling conditions in vivo it was crucial that the PTS protein concentrations be implemented at their high in vivo values. The model suggests a new interpretation of results hitherto not understood; in vivo, the major fraction of the PTS proteins may exist as complexes with other PTS proteins or boundary metabolites, whereas in vitro, the fraction of complexed proteins is much smaller.     

Determination of kinetic parameters from chemical relaxation data discrimination between coupled two-step binding reactions differing in stoichiometry

The chemical relaxation times of two different two-step equilibrium reactions, characterized by a 1:1 binding process followed by a subsequent rearrangement step and a stepwise 1:2 binding reaction, are analyzed for the purpose of qualitative model discrimination and quantitative determination of kinetic parameters. The equations describing the dependences of the two reciprocal relaxation times on suitable concentrations are given for both models in the general case as well as for four different limiting situations which are characterized by well separated relaxation times. The conditions corresponding to the limiting cases are expressed in terms of strong, weak and no coupling between the two partial equilibrium steps involved in both models. The coupling strength depends on the rate constants as well as on the total concentrations of the reactants. Criteria to discriminate between these two reaction models under defined limiting conditions are developed. In the general case, the product of both reciprocal relaxation times can be used to distinguish both models. If only one relaxation time can be resolved experimentally, it is possible under conditions described to determine only a reduced set of individual rate constants for most of the limiting cases considered. If both relaxation times are observed, all rate constants are determinable in the general case as well as in most of the limiting cases discussed.     

Estimation of kinetic parameters in a structured yeast model using regularisation.

In this work, a procedure for estimating kinetic parameters in biochemically structured models was developed. The approach is applicable when the structure of a kinetic model has been set up and the kinetic parameters should be estimated. The procedure consists of five steps. First, initial values were found in or calculated from literature. Hereafter using sensitivity analysis the most sensitive parameters were identified. In the third step physiological knowledge was combined with the parameter sensitivities to manually tune the most sensitive parameters. In step four, a global optimisation routine was applied for simultaneous estimation of the most sensitive parameters identified during the sensitivity analysis. Regularisation was included in the simultaneous estimation to reduce the effect of insensitive parameters. Finally, confidence intervals for the estimated parameters were calculated. This parameter estimation approach was demonstrated on a biochemically structured yeast model containing 11 reactions and 37 kinetic constants as a case study.     

Kinetic models and phenomenological analysis of passive lipid translocation in single-file

Passive movement of lipids through a membrane-embedded pore is analysed with kinetic equations of transport in single-file. The number of lipids arranged along the translocation coordinate in the pore is not limited in the calculations. The assumption is made that the energetic state of a pore is independent of the sequence of lipids contained in it. The results are valid for an arbitrary number of species with identical kinetic constants. It is shown that infinitely fast diffusion of one vacant site is equivalent to the filled pore approximation, which has been used here. We introduce the concept of non-strict single-file, which allows also for exchanges of neighbouring lipids inside the pore at specified rates. The model successfully simulates the redistribution of lipids between the monolayers of red blood cell plasma membranes under operation of an active aminophospholipid translocase. Kinetic equations are related to linear flux force relations. Phenomenological coefficients are expressed and analysed in terms of kinetic constants. Plausible kinetic pore model parameters are derived from comparison with a reference simulation of a thermodynamic model of the erythrocyte transmembrane lipid distribution. Mechanical forces due to differences in compressions of the lipid molecules between the monolayers are incorporated in kinetic rate constants. It is seen how the active inward transport of aminophospholipids causes an unsymmetrical passive redistribution of the other components due to mechanical effects and cross-coupling of fluxes.     

Allosteric interactions in phosphorylase B

An allosteric model of phosphorylase B is proposed based on the following assumptions. The enzyme consists of two sub-units and undergoes a concerted transition from the inactive T to the active R state. The binding of substrates, phosphate, and glycogen is regarded as exclusive, but the binding of the activator AMP is nonexclusive. The enzyme model is of the K, V type, i. e., the activator AMP is important, not only for the T-R transition and the substrates binding, but also for the formation of the active site. Therefore, it displays a big influence on the maximal reaction rate. Calculations based on this model lead to an equation containing 5 constants, which can be easily computed from kinetic data. All kinetic measurements fit the expressions derived from the model. Independent methods for the measurement of all the constants involved were developed. They are based on the study of binding of phosphorylase with the substrates and the activator. These measurements are in satisfactory agreement with the data obtained from enzyme kinetics.     

On true and apparent Michaelis constants in enzymology. III. Is it linear dependence between apparent michaelis constant and limiting rate and is it possible to determine the substrate constant value using this dependence?

The Slater-Bonner method which is used for graphic determination of substrate constant (Ks) by linear dependence of apparent Michaelis constant (Km(app)) on the limiting rate (V(app)) of enzyme-catalysed reactions with activator participation has been critically analysed. It has been shown that although it is possible to record the mechanisms of such reactions as a scheme similar to Michaelis-Menten model which allow to find correlation Km(app) and V(app) as equation Km(app) = Ks + V(app)/k1[E]0 ([E]0 is a total enzyme concentration, k1 is a rate constant of enzyme-substrate complex formation from free enzyme and substrate) in order to calculate Ks and individual rate constants (k1, k(-1)), but this approach for investigation of all reactions with activator participation ought not to be used. The above equation is not obeyed in general, it may be true for some mechanisms only or under certain ratios of kinetic parameters of enzyme-catalysed reactions.     

Simplified velocity equations for characterizing the partial inhibition or nonessential activation of bireactant enzymes

The steady state velocity equation for a bireactant enzyme in the presence of a partial inhibitor or nonessential activator, M, contains squared substrate concentration and higher-ordered M concentration terms. The equation is too complex to be useful in kinetic analyses. Simplification by the method of Cha (J. Biol. Chem. 243, 820 825 (1968)) eliminates squared substrate concentration terms, but retains higher-ordered terms in [M]. It is shown that if strict equilibrium is assumed between free E, M, and EM and for all but one other M-binding reaction, a velocity equation is obtained for an ordered bireactant enzyme that is first degree in all ligands in the absence of products. The equation is an approximation (because it was derived assuming only one M-binding reaction in the steady state), but it contains five inhibition (or activation) constants associated with M, all of which can be obtained by diagnostic replots and/or curve-fitting procedures. The equation also provides a framework for obtaining limiting constants (V'max, K'ia, K'mA, K'mB) that characterize the enzyme at saturating M. The same approach is applicable to an enzyme that catalyzes a steady state ping pong reaction.     

Mathematical modeling and analysis of monoclonal antibody production by hybridoma cells

An attempt has been made to mathematically describe and analyze monoclonal antibody (MAB) productivity of hybridoma cells, with particular emphasis on continuous cultures under unsteady-state conditions. A simple and unstructured general kinetic model that takes account of productivity loss during long-term cultivation, cell proliferation, and the effects of nutrients and toxic products is proposed. The model is verified with data of continuous culture from five different cell lines under a wide range of experimental conditions. Analysis of these results showed that for a reliable assessment of effects of different factors and for comparison of kinetic data on MAB production it is important to consider possible loss of MAB productivity, the time dependence of which can be modeled by an exponential function plus a constant term. Variations of nutrient concentration, particularly that of glucose, glutamine, and serum, can significantly alter MAB production under certain conditions. These effects can be described in terms of saturation kinetic and/or noncompetitive inhibition kinetics. (c) 1996 John Wiley & Sons, Inc.     

Determination of the intrinsic kinetic constants of immobilized glucose oxidase and catalase

Models of membrane systems containing immobilized glucose oxidase and catalase operating together or independently have been developed. A rotated disk electrode apparatus was employed with novel electrochemical operating conditions to experimentally determine mass transfer and intrinsic kinetic parameters of enzyme-containing membranes. The value of a mass transfer parameter that describes internal and external diffusion was first determined under conditions that do not permit the enzyme reactions. In a subsequent experiment with the reaction allowed, kinetic parameters corresponding to the intrinsic maximal velocity and Michaelis constants of the immobilized enzymes were estimated by regression analysis of data based on an appropriate two- or three- parameter model. It was found that immobilization reduced the maximal intrinsic velocity but had no detectable effect on the Michaelis constants. In all but one case- these methods for membrane characterization are nondestructive and can be used repeatedly on a given membrane. These techniques provide the means for quantitative comparisons of immobilization methods and make possible temporal studies of immobilized enzyme inactivation.     

Reduced kinetic models of facilitative transport

In spite of the highly complex structural dynamics of globular proteins, the processes mediated by them can usually be described in terms of relatively simple kinetic diagrams. How do complex proteins, characterized by undergoing transitions among a possibly very large number of intermediate states, exhibit functional properties that can be interpreted in terms of kinetic diagrams consisting of only a small number of states? One possible way of explaining this apparent contradiction is that, under some conditions, a reduction of the actual complete kinetic diagram that describes all of the macromolecular states and transitions takes place. In this work, we contribute with a formal basis to this interpretation, by generalizing the procedure of diagram reduction to the case of multicyclic kinetic diagrams. As an example, we apply the procedure to a complex kinetic model of facilitative transport. We develop Monte Carlo simulations to obtain the kinetic parameters of the complex model and we compare them with the ones analytically obtained from the reduced model. We confirm that, under some conditions, the kinetic behavior of the complex transporter is indistinguishable from the one of a four-state simple carrier model, derived from the former by diagram reduction. Besides introducing some novel methodological aspects, this work further contributes to the idea that, under many physiological and experimental conditions, a reduction occurs of the complete kinetic diagram that describes the dynamics of a globular protein.     

Estimating single-channel kinetic parameters from idealized patch-clamp data containing missed events.

We present here a maximal likelihood algorithm for estimating single-channel kinetic parameters from idealized patch-clamp data. The algorithm takes into account missed events caused by limited time resolution of the recording system. Assuming a fixed dead time, we derive an explicit expression for the corrected transition rate matrix by generalizing the theory of Roux and Sauve (1985, Biophys. J. 48:149-158) to the case of multiple conductance levels. We use a variable metric optimizer with analytical derivatives for rapidly maximizing the likelihood. The algorithm is applicable to data containing substates and multiple identical or nonidentical channels. It allows multiple data sets obtained under different experimental conditions, e.g., concentration, voltage, and force, to be fit simultaneously. It also permits a variety of constraints on rate constants and provides standard errors for all estimates of model parameters. The algorithm has been tested extensively on a variety of kinetic models with both simulated and experimental data. It is very efficient and robust; rate constants for a multistate model can often be extracted in a processing time of approximately 1 min, largely independent of the starting values.     

A delta pH clamp method for analysis of steady-state kinetics of photophosphorylation

An instrumental device is described which allows steady-state kinetic measurements of photophosphorylation at a desired proton gradient which can be maintained throughout the course of the experiment ('delta pH clamp'). This is achieved by electronic regulation of light intensity using the calibrated 9-aminoacridine fluorescence signal as sensor of the gradient. The instrument is suitable for determination of kinetic parameters of the proton-translocating ATPase in isolated envelope-free chloroplasts under defined conditions. At clamped delta pH, phosphorylation as a function of substrate concentration shows Michaelis-Menten kinetics. The true Michaelis constants and the dissociation constants for phosphate and ADP are reported. The Michaelis constants are not affected by the magnitude of the proton gradient in the investigated range. The significance of these results is discussed.     

The application of growth kinetics to the control of Eichhornia crassipes (Mart) solms. Through nutrient removal by mechanical harvesting

The growth of Eichhornia crassipes (water hyacinth) under nitrate-nitrogen limiting conditions can be explained by the Monod rectangular model. The kinetic constants maximum specific growth rate, half saturation constant and yield coefficient were determined under nitrate-nitrogen limiting conditions in nutrient culture at an air temperature of 25°C. The practical application of these constants for the possible control of water hyacinth through nutrient limitation is illustrated.     

Seeded isothermal batch crystallization of lysozyme

The kinetics of lysozyme crystallization under seeded isothermal batch conditions was followed by measurement of the decline in solution concentration versus time. Kinetics were measured for five different values of the seed crystal mass. The data were analyzed using a recently proposed mathematical model. For each seed mass, the model fit the kinetic data well. Growth rate constants determined using the model were approximately constant over a sixfold increase in the seed crystal mass, and fell well within the range of values reported in the literature, but obtained using entirely different experimental techniques. These results confirmed the utility of the proposed model. The proposed model can be used to analyze crystallization kinetics using absorbance measurements only, without the need to characterize the crystal size, thus avoiding the need for expensive laser light scattering and digital microscopy instrumentation. Thus, the model offers a low-cost straightforward method to analyze and simulate the effects of changes in operating parameters such as the seed crystal mass, solution volume, initial protein concentration, pH, temperature, salt concentration, and time.     

Kinetic and binding studies of Mn (II) and fructose 1,6-bisphosphate with rabbit liver hexosebisphosphatase.

The separate interaction of the substrate fructose 1,6-bisphosphate and a metal ion cofactor Mn2+ with neutral hexosebisphosphatase has been studied under equilibrium conditions at pH 7.5 with gel filtration and electron paramagnetic resonance measurements, respectively. Binding data for both ligands to the enzyme yielded nonlinear Scatchard plots that analyze in terms of four negatively cooperative binding sites per enzyme tetramer. Graphical estimates of the binding constants were refined by a computer searching procedure and nonlinear least squares analysis. These results are qualitatively similar to those obtained from binding studies involving teh alkaline enzyme, a modified form of hexosebisphosphatase whose pH optimum is in the alkaline pH region. Both forms of the enzyme enhance the proton relaxation rate of water protons by a factor of approximately 7 to 8 at 24 MHz, demonstrating similar metal ion environments. Teh activator Co(III)-EDTA did not affect Mn2+ binding to the neutral enzyme. In the presence of (alpha + beta)methyl-D-fructofuranoside 1,6-bisphosphate, however, two sets--each containing four Mn2+ binding sites--were observed per enzyme tetramer with loss of the negatively cooperative interaction. These results are viewed in terms of four noncatalytic and four catalytic Mn2+ binding sites. Parallel kinetic investigations were conducted on the neutral enzyme to determine specific activity as a function of Mn2+ and fructose 1,6-bisphosphate concentration. A pro-equilibrium sequential pathway model involving Mn2+-enzyme and the Mn2+-fructose 1,6-bisphosphate complex both as substrate and as an allosteric inhibitor satisfactorily fit the kinetic observations. All possible enzyme species were computed from the determined binding constants and grouped according to the number of moles of Mn2+-fructose 1,6-bisphosphate complex bound to the Mn2+-enzyme, and individual rate constants were calculated. The testing of other models and their failure to describe the kinetic observations are discussed.     

Mechanism of inhibitory effect of angiostatin on plasminogen activation by its physiologic activators

The influence of angiostatin K1-4.5--a fragment of the heavy chain of plasmin and a powerful inhibitor of angiogenesis--on kinetic parameters (k(Pg) and K(Pg)) of human Glu-plasminogen activation under the action of urokinase (uPA) not having affinity for fibrin and fibrin-specific tissue plasminogen activator (tPA) was investigated. Angiostatin does not affect the k(Pg) value, but increases the value K(Pg) urokinase plasminogen activation. A decrease in the k(Pg) value and an increase in the K(Pg) value were found for fibrin-stimulated plasminogen activation by tPA with increasing concentrations of angiostatin. The obtained results show that angiostatin is competitive inhibitor of the uPA activator activity, while it inhibits the activator activity of tPA by mixed type. Such an influence ofangiostatin on the kinetic constants ofthe urokinase plasminogen activation suggests that angiostatin dose dependent manner replaces plasminogen in the binary enzyme-substrate complex uPA-Pg. In case of fibrin-stimulated plasminogen activation by tPA, both zymogen and tPA are bound to fibrin with formation of the effective triple tPA-Pg-fibrin complex. Angiostatin replaces plasminogen both from the fibrin surface and from the enzyme-substrate tPA-Pg complex that leads to a decrease in k(Pg) and an increase in K(Pg) of plasminogen activation. Inhibition constants by angioststin (Ki) of plasminogen-activator activities of uPA and tPA determined by Dixon method were found to be 0.59 +/- 0.04 and 0.12 +/- 0.05 microM, respectively.     

Phenomenological theory of gel electrophoresis of protein-nucleic acid complexes

A phenomenological theory of gel electrophoresis is elaborated for protein-DNA complexes involving one, two, or three binding sites on the DNA molecule. The computed electrophoretic patterns simulate experimental patterns shown by both prokaryotic and eukaryotic systems. The mechanism whereby the electrophoretic protein-DNA ladder is generated upon titration of the operator with repressor is embodied in theory of mass transport coupled to reversible interactions under chemical kinetic control. In contrast to strong interactions (association constant greater than 10(12) M-1), patterns observed with weak complexes (K less than 10(10) M-1) could be simulated only by applying the cage effect, a model of which is formulated. Theoretical underpinning is provided for the electrophoretic estimation of equilibrium association constants, and requisite chemical kinetic conditions are elucidated for direct estimation of the rate constant for dissociation of the protein-DNA complex from gel patterns. The theory thus affords an experimenter with a means for determining the conditions required to render the gel retardation method a valid procedure for evaluating equilibrium constants and/or kinetic parameters for the particular protein-nucleic acid system under investigation. These several considerations apply not only to interactions of proteins with nucleic acids (DNA or RNA) but also to a wide range of macromolecular interactions involving peptides, drugs, and other ligands as well as large assemblies such as multienzyme complexes.     

Kinetics of glucose isomerization to fructose by immobilized glucose isomerase in the presence of substrate protection

The activity of immobilized glucose isomerase of Streptomyces murinus has been tested batchwise under different conditions in order to gather the related kinetic parameters necessary to optimize an immobilized enzyme column for the continuous production of high fructose corn syrup (HFCS). To this purpose, the Briggs-Haldane model incorporating an apparent first-order inactivation constant has been used with success. A comparison of the equilibrium constants and of the maximum theoretical conversion yields calculated at different temperatures with those estimated for the native enzyme demonstrates that the immobilization favours the transformation of glucose to fructose only at T?>?70?°C, as a possible consequence of a combined effect of catalysis and equilibrium thermodynamics enhancement. Enzyme inactivation has also been tested at different temperatures and sugar concentrations to evaluate the related kinetic parameters under different conditions of substrate protection.     

Inversion of Markov processes to determine rate constants from single-channel data.

The determination of rate constants from single-channel data can be very difficult, in part because the single-channel lifetime distributions commonly analyzed by experimenters often have a complicated mathematical relation to the channel gating mechanism. The standard treatment of channel gating as a Markov process leads to the prediction that lifetime distributions are exponential functions. As the number of states of a channel gating scheme increases, the number of exponential terms in the lifetime distribution increases, and the weights and decay constants of the lifetime distributions become progressively more complicated functions of the underlying rate constants. In the present study a mathematical strategy for inverting these functions is introduced in order to determine rate constants from single-channel lifetime distributions. This inversion is easy for channel gating schemes with two or fewer states of a given conductance, so the present study focuses on schemes with more states. The procedure is to derive explicit equations relating the parameters of the lifetime distribution to the rate constants of the scheme. Such equations can be derived using the equality between symmetric functions of eigenvalues of a matrix and sums over principle minors, as well as expressions for the moments, derivatives, and weights of a lifetime distribution. The rate constants are then obtained as roots to this system of equations. For a gating scheme with three sequential closed states and a single gateway state, exact analytical expressions were found for each rate constant in terms of the parameters of the three-exponential closed-time distribution. For several other gating schemes, systems of equations were found that could be solved numerically to obtain the rate constants. Lifetime distributions were shown to specify a unique set of real rate constants in sequential gating schemes with up to five closed or five open states. For kinetic schemes with multiple gating pathways, the analysis of simulated data revealed multiple solutions. These multiple solutions could be distinguished by examining two-dimensional probability density functions. The utility of the methods introduced here are demonstrated by analyzing published data on nicotinic acetylcholine receptors, GABA(A) receptors, and NMDA receptors.